Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
APPLIED MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-I and THREE Questions from SECTION-II, All questions carry EQUAL marks.
(iii)Use of Scientific Calculator is Allowed.
SECTION-1
Q. No. 1 (a) Prove that (A⃗ + B⃗) · (B⃗ + C⃗) × (C⃗ + A⃗) = 2 [A⃗ · (B⃗ × C⃗)]. (10)
(b) If A⃗ = (x – 3y)î + (y – 2x)ĵ, evaluate ∮_c A⃗ · dr⃗ where c is an ellipse x²/9 + y²/4 = 1 in the xy-plane traversed in the positive direction. (10)
Q. No. 2 (a) Determine the expression for divergence in orthogonal curvilinear coordinates. (10)
(b) Determine the unit vectors in spherical coordinate system. (10)
Q. No. 3 (a) A particle moves from rest at a distance “a” from a fixed point O where the acceleration at distance x is μx^(-5/3). Show that the time taken to arrive at O is given by an equation of the form t = A · (a^(4/3) / √μ), where A is a number. (10)
(b) Three forces P, Q, R acting at a point, are in equilibrium, and the angle between P and Q is double of the angle between P and R. Prove that R² = Q(Q – P). (10)
Q. No. 4 (a) AB and AC are similar uniform rods, of length a, smoothly joined at A. BD is a weightless bar, of length b, smoothly joined at B, and fastened at D to a smooth ring sliding on AC. The system is hung on a small smooth pin at A. Show that the rod AC makes with the vertical an angle tan⁻¹(b / (a + √(a² – b²))). (10)
(b) Find the centroid of the arc of the curve x^(2/3) + y^(2/3) = a^(2/3) lying in the first quadrant. (10)
Q. No. 5 (a) A hemispherical shell rests on a rough inclined plane whose angle of friction is λ. Show that the inclination of the plane base to the horizontal cannot be greater than sin⁻¹(2 sin λ). (10)
(b) A regular octahedron formed of twelve equal rods, each of weight w, freely jointed together is suspended from one corner. Show that the thrust in each horizontal rod is (3/2) √2 w. (10)
SECTION-II
Q. No. 6 (a) A particle is moving with uniform speed v along the curve x²y = a(x² + a² / √5). Show that its acceleration has the maximum value 10v² / 9a. (10)
(b) Discuss the motion of a particle moving in a straight line if it starts from rest at a distance a from a point O and moves with an acceleration equal to μ times its distance from O. (10)
Q. No. 7 (a) Prove that the force field F = (y² – 2xyz³)i + (3 + 2xy – x²y³)j + (6z³ – 3x²yz²)k is conservative, and determine its potential. (10)
(b) The components of velocity along and perpendicular to the radius vector form a fixed origin are respectively λr² and μθ². Find the polar equation of the path of the particle in terms of r and θ. (10)
Q. No. 8 (a) A particle is projected horizontally from the lowest point of a rough sphere of radius a. After describing an arc less than a quadrant, it returns and comes to rest at the lowest point. Show that the initial speed must be (sin α) √((2ag(1 + μ²)) / (1 – 2μ²)), Where μ is the coefficient of friction and aα is the arc through which the particle moves. (10)
(b) The law of force is Mu² and a particle is projected from or apse at distance a. Find the orbit when the velocity of the projection is √M / a². (10)
APPLIED MATHEMATICS, PAPER-II
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-I and THREE Questions from SECTION-II, All questions carry EQUAL marks.
SECTION-I
Q. No. 1. (a) Solve the initial value problem.
dy/dx + y/(2x) = x/y³, y(1) = 2 (10)
(b) Solve y” – 4y’ + 4y = e²ˣ (10)
Q. No. 2. Solve the following equations: (10)
(a) (1 – x²)(d²y/dx²) – 2x(dy/dx) + 2y = 0
(b) d³y/dx³ + dy/dx = cosec x (10)
Q. No. 3. (a) Classify the following: (5 each) (10)
(i) x²U_xx + (a² – y²)U_yy = 0 , -∞ < x < ∞, -a < y < a
(ii) U_xx – 6U_xy + 9U_yy + 3y = 0
(b) Solve (10)
∂²u/∂t² = ∂²u/∂x² 0 < x < 5
u(0,t) = u(5,t) = 0
u(x,0) = x²(x – 5)
u_t(x,0) = 0
SECTION-II
Q. No. 4. (a) Prove that if A_i and B_j are two first order tensors, then their product
A_i B_j (i,j = 1,2,3) is a second order tensor. (7)
(b) If ϕ(x₁, x₂, x₃) is a scalar point function then ∂ϕ/∂x_i are the components of a first
order tensor. (7)
(c) Find the invariant of the following second order tensor (6)
⎡2 4 -1⎤
⎢6 -7 10⎥
⎣3 -4 6⎦
Q. No. 5. (a) Verify that the transformation
x₁’ = 1/15 (5x₁ – 14x₂ + 2x₃)
x₂’ = -1/3 (2x₁ + x₂ + 2x₃)
x₃’ = 1/15 (10x₁ + 2x₂ – 11x₃)
is orthogonal and right handed. A vector field A⃗ is defined in the system
Ox₁x₂x₃ by A₁ = x₁², A₂ = x₂², A₃ = x₃²
Evaluate the components A’_i of the vector field in the new system Ox₁’x₂’x₃’. (7)
(b) Prove that any second order tensor A_ij can be written as the sum of a deviator
and an isotropic tensor. (7)
(c) If a_ij = a_ji are constants. Calculate:
∂² / (∂X_k ∂X_m) (a_ij X_i X_j) (6)
SECTION-III
Q. No. 6. (a) Find the real root of the equation by using Newton – Raphson’s method.
3x – cos x – 1 = 0 (10)
(b) Solve the following system of equations by Gauss-Seidel method.
Take initial approximation as x₁ = 0, x₂ = 0, x₃ = 0. Perform 3 iterations.
20x₁ + x₂ – 2x₃ = 17
3x₁ + 20x₂ – x₃ = -18
2x₁ – 3x₂ + 20x₃ = 25 (10)
Q. No. 7. (a) Find the real root of the equation x³ – 4x – 9 = 0 by Regular falsi method. Take
the interval of the root as (2,3) and perform 4 iterations. (10)
(b) Find a polynomial which passes through the following points:
x: -1 0 1 2
f(x): 2 1 2 5 (10)
Q. No. 8. (a) Use the Lagrange’s Interpolation formula to find the value f(12) if the values of x
and f(x) are given below
| x | 5 | 7 | 11 | 13 |
|—–|—–|—–|——|——|
| f(x)| 150 | 392 | 1452 | 2366 | (10)
(b) Evaluate ∫₀¹ x√(1 + x²) dx using 1/3 Simpson’s rule and trapezoidal rule for n = 6 (10)
*************
×